In this paper, we discuss some general features of two-body decay kinematics and derive a new set of equations describing boundaries of the physical region in decays. These features via a graphical representation and implementation of selection criteria provide an efficient tool for the background suppression in experimental searches for conventional and exotic (pentaquark, etc.) resonance states.

1. Introduction

In the experimental search for a resonance production in particle collisions, momenta of secondary particles are combined to form the momentum of a resonance candidate. In high-energy reactions with high multiplicity of final states, this method of resonance reconstruction may cause a large combinatorial background which makes searches for new narrow states in invariant-mass distributions to be rather difficult. This was the case with the top-quark discovery, searches for new charmonium states or pentaquark searches. For reconstruction of the signal with a sufficient statistical significance, any additional physical information about the resonance and its decay properties (lifetime, the mass of secondary particles, etc.) helps considerably to reduce the background. For instance, the presence in an event a well-separated secondary vertex allows us to reconstruct and mesons from the -meson family with a very low background. One such example is the study of meson resonances in the final state in deep inelastic scattering at HERA [1]. Moreover, if masses of decay products are known and significantly differ from each other, this information also has to be used to reduce the background. This paper is a discussion of these issues.

Below only two-body decay modes are considered, and for each final state particle a set of kinematic equations describing boundaries of the physical region is given. Proposed selection criteria are based on the features of each physical region [2] and proved to be the efficient tool for experimental reconstruction both the conventional as well as exotic resonances [3, 4].

2. Two-Particle Decay

Let us consider a two-particle decay, , of a resonance with a mass and a momentum in the laboratory frame. Masses of the decay products are denoted by and . At the rest frame of the particles, and are flying in opposite directions with the momentum [5, 6]: In the laboratory frame, the absolute momenta and of the particles and depend on the relative orientation of the rest frame vectors and with respect to the boost vector. We shall consider only Lorentz boosts along the momentum and denote by the polar angle between and the direction given by . In that case, .

The energy and momentum components in both frames are related via [5, 6]: For a boost along , the boost parameters are and therefore If in an experiment there is no possibility to determine a particle type corresponding to a given charged track, then only information about the particle momentum (2.5) is used. In the opposite case, when the particle type can be identified, for instance, by ionization and time of flight, one may incorporate this information and in addition use (2.2). Below, these cases are considered separately.

2.1. Particles without Identification

The boundaries of the physical regions of the particle on the plane are easy to obtain with the use of (2.5). For a given , reaches the upper limit at : when the lower limit at : The equations similar to (2.6)-(2.7) are valid for the particle too. At the resonance rest frame, (2.7) demonstrates an interesting feature of the momentum of the particle flying backward. With increasing , the momentum () first decreases, at it reaches the zero value and only at lager starts to increase. When plotted on the momentum plane, (2.6)-(2.7) select a band-like physical region (-band). For secondary particles with equal masses, , these -bands are fully overlapping. However, if for instance, , the -bands overlap only partially or even separate out at The last equation follows from the condition . In (2.8), the expression under the square root is positive only if Thus, for the physical regions of the particles and do not overlap if and .

We shall now consider a few particular applications of (2.6)–(2.10). According to the PDG [7] mass values, only resonances with mass close to the threshold value satisfy the condition (2.10). Some of them are listed in Table 1, where and values were calculated with use of (2.1) and (2.8), respectively.

Figures 1(a), 1(b), and 1(c) shows -bands of decay products for some resonances listed in Table 1. For all of them always at , and significantly larger . We will refer to the momentum condition (2.11) as a -selector. Thus, for resonances of the type in Table 1, fulfillment of the condition (2.11) allows an assignment to the particle of the mass , that is, we identify the particle . The -selector (2.11) is a powerful condition for the background suppression.

In the two-body decay modes, one meets with a higher incidence the situation shown in Figure 1(d). For these decays the conditions (2.9)-(2.10) are not fulfilled, and at any the phase space bands remain overlapping. Nevertheless, if , one can demand fulfillment of (2.11) for the particle in order to assign it the mass . In course of the resonance search, the condition (2.11) rejects not only a significant part of the background, but also some fraction of the signal combinations. To estimate the efficiency of the -selector we proceed in the following way.

On plane, the condition (2.11) is valid up to the line defined by the equation: For decays the last equality is shown in Figure 1(d) by the dashed line. The true is a combination of a kaon from the region above the line (2.12) with a pion below it and vice versa. From (2.12) and (2.5), we find the variation of along the line (2.12): At the rest frame of , (2.11) is equivalent to the exclusion of the region . For unpolarized particles the distribution of is uniform, and we define the efficiency of the -selector (2.11) as With the definition (2.14), we get , when the conditions (2.9)-(2.10) are fulfilled, and for decays with (for real data , see discussion in Section 3). As another example, in Figure 4 by the dashed line is shown the evolution of with in decays. grows because with the increase of the overlap of -bands decreases.

In the limit large , the expression for the efficiency can be simplified: The last equation is a good approximation only if . Equations (2.15)-(2.16) confirm the result we have already seen in Figure 4(a), where is independent of at large . On the other hand, for fixed values of and , decreases with increasing , the mass of the resonance candidate. Thus, the higher the invariant mass of a two-particle combination, the more strongly (2.11) suppresses that part of the mass spectrum.

2.2. Identified Particles

Accounting for the particle masses transforms (2.6)-(2.7) into and at low leads to a “repulsion” between the phase space -bands on the energy () plane (Figure 2). For all resonances with (see Table 1): independent of . This is not always true for the background combinations. Therefore, if applied, the condition (2.18) suppresses the background even more strongly than (2.11). We will refer to the energy condition (2.18) as an -selector.

As in the previous section, if the masses of the secondary particles do not differ significantly, at greater than the -bands start overlapping in the way shown in Figure 2(d). The loss of signal combinations by demanding (2.18) for resonance candidates we estimate again with (2.14). In the case under consideration, is a solution of the equation: Thus,

In the limit large , with (2.21) we again recover (2.15)-(2.16). The evolution of in decays is shown in Figure 4(a) by the full line. at and drops down up to the value (2.15), , with an increase in .

3. Remark about Reconstruction

Charged particles are tracked in a tracking detector (TD). The resolution of the transverse momentum of a track traversing the TD is parametrized by , with being the track transverse momentum (in GeV). The coefficients , and characterize the resolution of the TD. Usually, to increase the momentum resolution, only tracks with –0.15 GeV are selected. That cut, as shown in Figure 1(c), makes impossible the reconstruction of mesons with momenta lower than 1.8–2.0 GeV. The -band of mesons is very narrow and grows rather slowly with . Tracks with momenta greater than belong to the background. This property can be used to suppress the background in the distribution on the mass difference, , by imposing on the momentum of the soft pion , the following cut, where –0.3 GeV is a some shift from the pion -band. In the decay , the rest frame momentum is small,  GeV, and the momentum can be estimated with (2.6) by means of the reconstructed momentum: Thus, from (3.2) and (2.6) one get with (3.1): Instead of (3.3), it is possible to apply another, less-strict cut: Here, is the right-hand edge of the momentum range of the candidates, .

4. Pentaquark States

Now we apply the results of the previous sections to new resonances predicted [8, 9] by Diakonov, Petrov, and Polyakov in the framework of the chiral soliton model and detected both in the formation type [4, 1018] and in the production type [3] experiments. This is so-called five-quark states (pentaquarks). For a review on pentaquark phenomenology, see [19].


With the mass value predicted for ,  GeV, and the masses (in calculations were used mass values averaged over the isomultiplet) of decay products,  GeV and  GeV, the condition (2.9) is not satisfied. That implies the overlap of the -bands at all (Figure 3(a)) as well as the overlap of the -bands at  GeV (Figure 3(b)). The overlapping is not strong and both selectors (2.11) and (2.18) work with high efficiency (see Figure 4). The - and -selectors [2] were successfully applied in searches for the in the production-type [3, 20, 21] and the formation-type [4, 17] experiments.

The situation with the particle searches is rather complicated and confusing, especially when similar experiments have different conclusions on a signal existence. To be more specific, let us remind that two similar collider experiments at DESY (Hamburg) [3, 23] and two similar fixed-target experiments at IHEP (Protvino) [4, 24] do not agree on the existence of an exotic state with the mass close to 1530 MeV, which may be interpreted as a strange pentaquark state [8]. While the ZEUS [3] and SVD-2 [4] experiments claim the observation of a narrow peak near 1530 MeV in the decay mode , the H1 and SPHINX collaborations do not see it. The inconsistency of the experimental situation with searches of baryon has risen even more, after the recent public polemic on results obtained with the CLAS detector at Thomas Jefferson National Accelerator Facility [25, 26]. All this discussion is beyond the scope of the present paper and can be found elsewhere [27, 28].


A baryonic state is a weakly decaying particle. In a cascade decay, the vertex of the decay is separated from the primary vertex, similarly to decays. This property is used to suppress combinations which do not result from a decay and helps to reconstruct the candidate and its invariant mass.

In spite of the large mass asymmetry, , the conditions (2.9)-(2.10) are not fulfilled in decays, and the pictures of the - and -bands shown in Figures 3(c) and 3(d) are very similar to those in decays. At low momenta, the efficiency of the -selector is 100. The -bands start overlapping at  GeV. Thus, the efficiency of the -selector is no worse than 84 (Figure 4).


The decays strongly and exclusively at the primary event vertex. The decay modes are dominant. Therefore, after the reconstruction of , the use of the - or -selector allows reconstruction of the with high efficiency (see Figures 1(b) and 2(b)).

Figures 3(e) and 3(f) show the - and -bands in decays. The -bands slightly overlap at low and diverge at  GeV. The -bands are totally separated. Thus, in that decay mode, the - and -selectors work with 100 efficiency at high momenta (Figure 4).

4.4. or ?

The NA49 collaboration provided an evidence for the existence in the decay mode a narrow baryonic state with the mass of  GeV [22]. This state is considered as a candidate for the exotic pentaquark state . The reported mass value is much lower as predicted in [8]. The last developments in the theory of pentaquark states did not exclude the lower mass for the [9]. In Figure 5 are shown the -bands and the efficiency of both selectors. They are much the similar to those in Figures 3(e) and 4(d). The picture of the -bands is also similar to Figure 3(f). Thus, the -selector will not suppress the signal but will suppress the background at higher masses.

We have to note that there are arguments [29] that the result of the NA49 collaboration perhaps is inconsistent with data collected over the past decades.

5. Conclusions

Kinematics of the two-body decay, , has been analyzed in terms of equations describing boundaries of the physical region in decays and the graphical phase space - and -bands. The efficiency of the - and -selectors was estimated with the use of (2.14). On the basis of many examples, in particular, the exotic antidecuplet baryons (pentaquark states), it has been demonstrated that for the selection rules and can be with a high efficiency applied for the signal reconstruction from many resonances.


This work is dedicated to the memory of Professor Pavel F. Ermolov (30.06.1932-14.05.2008), a prominent physicist, the organizer of Department of Experimental High Energy Physics, SINP MSU, the main “driving force” of the SVD experiment and a wonderful person.


The author is indebted to P. Bussey and S. Chekanov for comments and grateful to other colleagues from the ZEUS collaboration for useful discussions. Special thanks are due to P. F. Ermolov and A. Kubarovsky for discussions of preliminary results of the SVD collaboration.